\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,amsthm,enumerate,fancyhdr,hyperref,parskip,lastpage}
\usepackage[left=2.6cm, right=2.6cm, top=1.5cm, includehead, includefoot]{geometry}

%% header
\pagestyle{fancy}
\fancyhead[L]{\bf\large PMATH 445/745: Representation Theory\\ Assignment 3}
\fancyhead[R]{\bf\large Winter 2025  \\Due Friday, Jan 31 at 7:00 PM}
%\fancyfoot[C]{Page \thepage\ of \pageref*{LastPage}}
\setlength{\headheight}{35pt}

%% macros
\newcommand{\C}{\mathbb{C}}
\newcommand{\Fun}{\mathcal{F}}
\newcommand{\Hom}{\mathrm{Hom}}
\newcommand{\triv}{\mathrm{triv}}
\newcommand{\sgn}{\mathrm{sgn}}
\newcommand{\std}{\mathrm{std}}
\newcommand{\reg}{\mathrm{reg}}
\newcommand{\Irr}{\mathrm{Irr}}

\begin{document}
	\noindent In the problems below, $G$ is a finite group and all representations are finite-dimensional. You may assume $F=\C$.
	
	\begin{enumerate}[{\bf Q1.}]
		\item Determine the isotypic decompositions of the following $\C S_3$-modules. Write the decompositions in terms of $\Irr_\C(S_3) = \{V_\triv, V_\sgn, V_\std\}$.
		\begin{enumerate}
			\item $V_\reg$.
			\item $(V_\std)^*$.
			\item $V_\std \otimes V_\std$.
		\end{enumerate}
		
		\hrulefill
		\medskip
		
		\textit{Solution:}
		
		\begin{enumerate}
			\item 
			\item 
			\item
		\end{enumerate}
		
		\newpage
		
		\item Let $V=\C^2$ be the representation of $C_4=\langle a\rangle$ in which $a$ acts as a 90-degree counterclockwise rotation. Determine the isotypic decomposition of $W=\Hom(V,V)$ in terms of $\Irr_\C(C_4) = \{\chi_0, \chi_1,\chi_2,\chi_3\}$, where $\chi_j(a) = \exp(2\pi i j/4)$. [You should do this without writing down a single matrix!]
		
		\hrulefill
		\medskip
		
		\textit{Solution:}
		
		\newpage
		
		\item Let $V$ be an $FG$-module. A bilinear form $B\colon V\times V\to F$ is said to be \textbf{$G$-invariant} if
		\[ B(gv,gw) = B(v,w) \ \text{for all } g\in G \text{ and } v,w\in V. \]
		Assume that $V$ is irreducible and $F$ is algebraically closed.
		\begin{enumerate}
			\item Prove that if $B\colon V\times V\to F$ is a $G$-invariant bilinear form then there exists a scalar $c\in F$ such that $B(v,w) = cB(w,v)$ for all $v,w\in V$. [\textbf{Hint:} Use $B$ to construct two maps $V\to V^*$.]
			
			\item Assume that there exists a non-zero $G$-invariant bilinear form $B\colon V\times V\to F$ and choose a scalar $c$ as in part (a). Prove that if $B'$ is another $G$-invariant bilinear form then $B'(v,w) = c B'(w,v)$ for all $v,w\in V$ and with the same scalar $c$. Thus, we may write $c_V$ for the scalar $c$ since it is independent of the choice of $B\neq0$. Note that $c_V\neq0$.
			
			In the case where the only $G$-invariant bilinear form is the zero form, we set $c_V=0$.
			
			
			%% OLD (b)
			%\item The scalar $c\in F$ in part (a) depends only on $V$ and not on the bilinear form $B$. That is, prove that if $B'$ is another $G$-invariant bilinear form and if $B(v,w)=cB(w,v)$ for all $v,w\in V$ then $B'(v,w)=c B'(w,v)$ for all $v,w\in V$. Thus, we may write $c_V$ for this scalar $c$.
			
			\item Prove that $c_V \in \{0,\pm1\}$ and $c_V = 0$ if and only if $V\not\cong V^*$ (as representations).
		\end{enumerate}
		
		\hrulefill
		\medskip
		
		\textit{Solution:}
		
		\begin{enumerate}
			\item 
			\item 
			\item
		\end{enumerate}
		
	\end{enumerate}
\end{document}